Integrated Math 1 Module 2 Systems of Equations and...

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© 2012 Mathematics Vision Project | MVP In partnership with the Utah State Office of Education

Licensed under the Creative Commons Attribution‐NonCommercial‐ShareAlike 3.0 Unported license

Integrated Math 1 Module 2

Systems of Equations and Inequalities

Adapted from

The Mathematics Vision Project: Scott Hendrickson, Joleigh Honey, Barbara Kuehl,

Travis Lemon, Janet Sutorius www.mathematicsvisionproject.org

In partnership with the Utah State Office of Education

http://www.mathematicsvisionproject.org/

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SDUHSD Math 1 College Prep

Module 2 Overview Prerequisite Concepts & Skills:

Operations with integers, fractions, decimals and variable expressions Solve multi step equations and inequalities Graph linear equations in slope-intercept and standard form Evaluate expressions using the order of operations Write linear equations and inequalities to represent a context

Summary of the Concepts & Skills in Module 2:

Reinforce group roles and communication skills (orally & written) CC Standards of Math Practices through daily tasks Write linear equations and inequalities to represent a set of constraints Use graphs to solve systems of equations and inequalities Use technology (Graphing Calculators/Desmos) to graph linear functions and determine the most

appropriate window to use. Solve systems of equations algebraically Identify types of solutions of a system of linear equations including one solution, no solution, or infinitely

many solutions Interpret solutions of systems in the context of a situation. Determine if a given point is a solution to an equation, inequality, or system of equations Write an objective function to determine the optimal solution for a situation Identify corner points of a feasible region of the graph of a system of inequalities algebraically and

graphically Understand that the optimal solution for linear programming problems is always on the boundary of the

feasible region Content Standards and Standards of Mathematical Practice Covered:

Content Standards: A.CED.2, A.CED.3, A.CED.4, A.REI.5, A.REI.6, A.REI. 10, A.REI.12, A.SSE.1, N.Q.1, N.Q.2, F.LE.1b, F.LE.5

Standards of Mathematical Practice: 1. Make sense of problems & persevere in solving them. 2. Reason abstractly & quantitatively 3. Construct viable arguments & critique the reasoning of others 4. Model with mathematics 5. Use appropriate tools strategically 6. Attend to precision 7. Look for and make use of structure 8. Look for and express regularity in repeated reasoning

Materials:

Graphing utility (Graphing calculator/Desmos) Black line master of the “Module 1 Review” (Enough for one problem per group) Colored pens/pencils Graph paper

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Module 2 Vocabulary: System of Equations/Inequalities Constraint Solution region Feasible region Objective function Optimal solution Point of intersection Boundary of the solution region/feasible region Inconsistent solution for a system of equations Dependent solution for a system of equations At least More than/No more than Solid/Dotted line

Concepts Used In the Next Module:

Sequences - arithmetic, geometric, and other Represent sequences using dot/tile diagrams, context, tables, graphs, and equations (recursive and

explicit) Arithmetic and geometric mean Identify a common difference/ratio of arithmetic and geometric sequences

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Module 2 – Systems of Equations and Inequalities 2.1 An introduction to representing constraints with systems of inequalities and Writing and graphing linear inequalities in two variables (A.CED.2, A.CED.3, A.REI.12) Warm Up: Pet Sitters‐ A Develop Understanding Task Classroom Task: Too Big or Not Too Big, That is the Question ‐ A Solidify Understanding Task Ready, Set, Go Homework: Systems 2.1 2.2 Writing and solving equations in two variables (A.CED.2, A.CED.4) Warm Up: Name That Equation Classroom Task: Some of One, None of the Other– A Solidify Understanding Task Ready, Set, Go Homework: Systems 2.2 2.3 Writing and graphing inequalities in two variables to represent constraints (A.CED.2, A.CED.3, A.REI.12) Warm Up: Profiting from Pictures Classroom Task: Tamara’s Trucks – A Solidify Understanding Task Ready, Set, Go Homework: Systems 2.3 2.4 Writing and graphing inequalities in two variables to represent constraints and Graphing the solution set to a linear system of inequalities (A.CED.2, A.CED.3, A.REI.12) Warm Up: Pampering and Feeding Time – A Practice Understanding Task Classroom Task: All for One, One for All – A Solidify Understanding Task Ready, Set, Go Homework: Systems 2.4 2.5 Solving systems of linear equations in two variables (A.REI.6) Warm Up: Tickets Tickets Classroom Task: Get to the Point – A Solidify Understanding Task Ready, Set, Go Homework: Systems 2.5 2.6 An introduction to solving systems of linear equations by elimination (A.REI.5, A.REI.6) Warm Up: Graphing a linear inequality from a context Classroom Task: Shopping for Cats and Dogs – A Develop Understanding Task Ready, Set, Go Homework: Systems 2.6 2.7 Solving systems of linear inequalities representing constraints Warm Up: Profiting Classroom Task: Food for Fido and Fluffy – A Solidify Understanding Task (A.CED.3) Set, Go Homework: Systems 2.7 2.8 Working with systems of linear equations, including inconsistent and dependent systems (A.REI.6) Warm Up: In Or Out Classroom Task: Taken Out of Context – A Practice Understanding Task Set, Go Homework: Systems 2.8 2.9 Using systems of linear equations and inequalities in a modeling context (High School Modeling Standard) Warm Up: Profiting from Hens and Abacuses Classroom Task: Pet Sitters Revisited – A Develop Understanding Task Ready, Go Homework: Systems 2.9

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2.10 Module 2 Review Warm Up: Reflecting on Carlos and Clarita Classroom Task: Linear Programming Carousel Go Homework: Systems 2.10

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2.1 Warm Up: Pet Sitters A Develop Understanding Task The Martinez twins, Carlos and Clarita, are trying to find a way to make money during summer vacation. When they overhear their aunt complaining about how difficult it is to find someone to care for her pets while she will be away on a trip, Carlos and Clarita know they have found the perfect solution. Not only do they have a large, unused storage shed on their property where they can house animals, they also have a spacious fenced backyard where the pets can play. Carlos and Clarita are making a list of some of the issues they need to consider as part of their business plan to care for cats and dogs while their owners are on vacation.

Space: Cat pens will require 6 ft2 of space, while dog runs require 24 ft2. Carlos and Clarita have up to 360 ft2 available in the storage shed for pens and runs, while still leaving enough room to move around the cages.

Startup Costs: Carlos and Clarita have at most $1280 to purchase cat pens and dog runs. It will cost $32 for each cat pen and $80 for each dog run.

Of course, Carlos and Clarita want to make as much money as possible from their business, so they are trying to determine how many of each type of pet they should plan to accommodate. They plan to charge $8 per day for boarding each cat and $20 per day for each dog. After surveying the community regarding the pet boarding needs, Carlos and Clarita are confident that they can keep all of their boarding spaces filled for the summer. So the question is: How many of each type of pet should they prepare for in order to get the highest daily income? Their dad has suggested the same number of each, perhaps 12 cats and 12 dogs. Carlos thinks they should plan for more dogs, since they can charge more. Clarita thinks they should plan for more cats since they take less space and time, and therefore they can board more.

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2.1 Resource Page What do you think?

Number of Dogs

Number of Cats

Space Used Start-up Costs Revenue

What recommendations would you give to Carlos and Clarita, and what argument would you use to convince them that your recommendation is reasonable?

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2.1 Too Big or Not Too Big, That is the Question A Solidify Understanding Task As Carlos is considering the amount of money available for purchasing cat pens and dog runs (see below) he realizes that his father’s suggestion of boarding “the same number of each, perhaps 12 cats and 12 dogs” is too big. Why?


1. Find at least 5 more combinations of cats and dogs that would be “too big” based on this Startup Cost

constraint. Plot each of these combinations as points on a coordinate grid using the same color or symbol for each point.

2. Find at least 5 combinations of cats and dogs that would not be “too big” based on this Startup Cost constraint.

Plot each of these combinations as points on a coordinate grid using a different color or symbol for the points than you used in #1.

3. Find at least 5 combinations of cats and dogs that would be “just right” based on this Startup Cost constraint.

That is, find combinations of cat pens and dog runs that would cost exactly $1280. Plot each of these combinations as points on a coordinate grid using a third color or symbol.

4. What do you notice about these three different collections of points? 5. Write an equation for the line that passes through the points representing combinations of cat pens and dog

runs that cost exactly $1280 (Question 3). What does the slope of this line represent?

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Carlos and Clarita don’t have to spend all of their money on cat pens and dog runs, unless it will help them maximize their profit. 6. Shade all of the points on your coordinate grid that satisfy the Startup Costs constraint. 7. Write a mathematical rule to represent the points shaded in #6. That is, write an inequality whose solution

set is the collection of points that satisfy the Startup Costs constraint.

In addition to Startup Costs, Carlos needs to consider how much space he has available, based on the following:


8. Write an inequality to represent the solution set for the space constraint. Shade the solution set for this

inequality on a different coordinate grid.

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2.1 Resource Page Plot points that represent the following conditions in different colors/symbols based on the Start-up Cost constraint:

Amount of dogs and cats that are “too big.” Amount of dogs and cats that are “too small.” Amount of dogs and cats that are “just right.”

Write an inequality whose solution set is the collection of points that satisfy the Start-up Cost constraint:

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2.1 Resource Page Plot points that represent the following conditions in different colors/symbols based on the Space constraint:

Amount of dogs and cats that are “too big.” Amount of dogs and cats that are “too small.” Amount of dogs and cats that are “just right.”

Write an inequality whose solution set is the collection of points that satisfy the Space constraint:

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Name: Systems 2.1

Ready, Set, Go!

Ready Topic: Determine if given value is a solution and solve systems of equations 1. Graph both equations on the same axes. Then determine which ordered pair is a solution to the system of

linear equations. and

a. b. c.

2. Solve the following system by graphing. Check the solution by evaluating both equations at the point of

intersection. and

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Topic: Determining if given values are solutions to an equation 3. Identify which of the given points are solutions to the following linear equation. Then graph the equation.

a.

b.

c.

d.

Find the value that will make each ordered pair a solution to the given equations. Then graph each equation. 4.

a. b. c.

5.

a. b. c. ( )

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Set Topic: Determining possible solutions 6. a. A theater wants to take in at least $2000 for a certain matinee. Children’s tickets cost $5 each and adult

tickets cost $10 each. The theater can seat up to 350 people. Find five combinations of children and adult tickets that will make their goal.

b. Plot those points on the graph below, then find and plot five additional combinations of children and adult

tickets that will make their goal.

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Topic: Graph linear inequalities 7.

a. Solve for y. b. Now imagine that your inequality is an equation. In other words, your

solution will say , instead of , or . With the equal sign, it should be the equation of a line. Graph your equation.

c. Find the y–intercept. d. Find the slope. e. Select a point that is above the line.

Replace the x and y – values in the inequality: Is the inequality still true?

f. Select a point that is below the line.

Replace the x and y – values in the inequality: Is the inequality still true?

g. Explain which side of the line should be shaded.

h. Decide whether the line should be solid or dotted. Justify your decision.

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Go Topic: Graphing linear equations and determining if a given value is a solution Graph each equation below, then determine if the point is a solution to the equation. Name two additional points that are solutions to the equation and show these points on the graph.

8. 9.

Topic: Solving inequalities Follow the directions for each problem below. (Show your work!) 10.

a. Solve for x. Then graph the solution on the number line.

b. Select an x-value from your graph of the solution of the inequality. Replace x in the original inequality

with your chosen value. Does the inequality hold true?

c. Select an x-value that is outside of the solution set on your graph. Replace x in the original inequality with your chosen value. Does the inequality still hold true?

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Topic: Graph linear inequalities Graph the following inequalities on the coordinate plane. Name one point that is a solution to the inequality and one point that is not a solution. Show algebraically and graphically that your points are correct.

11.

12.

13.

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2.2 Warm Up: Name That Equation #2 Creating an equation from a context The airplane Mrs. Samaniego flew in cruised at an altitude of 30,000 feet. The descent into the Reno airport, which is at 1500 feet of elevation, took 25 minutes. a. What was the starting point (vertical axis or y-intercept)?

b. What was the rate of change (slope)? c. Write an equation modeling the altitude (a) of the airplane when it descends for (t) minutes, assuming the

plane descends at the same rate all the way down. d. Use your equation to predict the altitude of the airplane when it had descended for 20 minutes.

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2.2 Some of One, None of the Other A Solidify Understanding Task Carlos and Clarita are comparing strategies for writing equations of the boundary lines for the “Pet Sitter” constraints. They are discussing their work on the Space constraint.


Carlos’ Method: “I made a table. If I don’t have any dogs, then I have room for 60 cats. If I use some of the space

for 1 dog, then I can have 56 cats. With 2 dogs, I can boards 52 cats. For each additional dog, I can board 4 fewer cats. From my table I know the yintercept of my line is 60 and the slope is , so my equation is .”

Clarita’s Method: “I let x represent the number of dogs, and y the number of cats. Since dog runs require 24 ft2,

24x represents the amount of space used by dogs. Since cat pens require 6 ft2, 6y represents the space used by cats. So my equation is .”

1. Since both equations represent the same information, they must be equivalent to each other.

a. Show the steps you could use to transform Clarita’s equation into Carlos’ equation. Justify each step.

b. Show the steps you could use to transform Carlos’ equation into Clarita’s. Justify each step.


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2. Use both Carlos’ and Clarita’s methods to write the equation of the boundary line for the Startup Costs constraint.


a. Carlos’s Method:

b. Clarita’s Method: 3. Show the steps you could use to transform Clarita’s Startup Costs equation into Carlos’ equation. Justify each

step.

4. Show the steps you could use to transform Carlos’ Startup Costs equation into Clarita’s. Justify each step. In addition to writing an equation of the boundary lines, Carlos and Clarita need to graph their lines on a coordinate grid. Carlos’ equations are written in slopeintercept form (ex: ). Clarita’s equations are written in standard form (ex: ). Both forms are ways of writing linear equations. Both Carlos and Clarita know they only need to plot two points in order to graph a line. 5. Carlos’ strategy: Explain how Carlos might use his slope‐intercept form, , to plot two points on

his line? Plot the points on the provided resource page. 6. Clarita’s strategy: Explain how Clarita might use her standard form, , to plot two points on

her line? (Clarita is really clever, so she looks for the two easiest points she can find.)

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2.2 Resource Page Graph the linear equations for the Start-up Cost constraint and the Space constraint on the grid below.

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Name: Systems 2.2

Ready, Set, Go!

Ready Topic: Determining points that satisfy equations and solving systems of equations Three points are given. Each point is a solution to at least one of the equations. Find the point that satisfies both equations. (This is the solution to the system) Justify that the point is a solution to both equations and that the others are not.

1. {

2. {

a. ) a.

b. b.

c. c.

3. {

4. {

a. a.

b. b.

c. c.

Set Topic: Graphing linear equations from standard form using intercepts Graph the following equations by finding the intercepts. 5. 6.

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7. 8.

Go Topic: Solving equations containing fractions and decimals. Solve the following equations. Check your solutions.

9.

10.

11.

12.

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2.3 Warm Up Profiting from Pictures Brian is an artist who specializes in abstract art. He is getting ready for a big art expo next month. Brian specializes in both acrylic paint and pastels. Each type of pictures takes him about the same amount of time to paint. He thinks he has time to make a total of at most 21 pictures. The materials for each acrylic painting will cost $20. The materials for each pastel art piece will cost $4. He has $180 to spend on materials. He makes a profit of $50 on each pastel and a profit of $120 on each acrylic painting. 1. Express Brian’s constraints as inequalities. Use a to represent the number of acrylic paintings and p to

represent the number of pastels. 2. Make a graph that shows Brian’s feasible region. The graph should show all the combinations of acrylic

paintings and pastels that satisfy his constraints.

3. For at least five points on your graph, find the profit that Brian would make for that combination.

4. Write an algebraic expression to represent Brian’s profit in terms of a and p.

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2.3 Tamara’s Trucks A Solidify Understanding Task Tamara is a long distance truck driver, but she has found that she can’t make a living only driving cross country. So she has bought a smaller semi-truck from Mike. He agrees to help her set up her business. Mike has two customers he no longer has time to serve and suggest that Tamara work for them delivering their merchandise. One is Heidi’s Hen Houses and the other is Amy’s Abacuses. Heidi’s Hen Houses packages its product in containers that weigh 40 pounds and are 3 cubic feet in volume. Mike has been charging $4 a container. Amy’s Abacuses packages its material in containers that weigh 30 pounds and are 4 cubic feet in volume. Mike has been charging $2.50 a container. Mike told Tamara that the plane can hold a maximum of 2800 cubic feet of material and that the maximum weight it can carry is 31,500 pounds. Heidi’s Hen Houses and Amy’s Abacuses have both told Tamara that they can give her as much business as she can handle. Of course, she wants to maximize the money she gets per trip so she can spend more time driving cross country. 1. Write down and organize the key information.

2. Define the variables. 3. Write the constraints. Include a brief description of each.

4. Write an expression using your variables to show how much Tamara will earn for transporting containers for

Heidi’s Hen Houses and Amy’s Abacuses. 5. Come up with four loads for Tamara that fit the constraints and figure out how much she will earn for each. 6. Come up with four loads for Tamara that fit one of the constraints, but not the other.

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7. Label your axes showing the number of containers of Heidi’s Hen Houses on the x-axis and the number of containers of Amy’s Abacuses on the y-axis. Graph the constraints shade the feasible region.

8. Find the coordinates of the corners of the feasible. Identify the intersection of the two lines and interpret the

meaning of this point in the context of the problem.

9. What would be Tamara’s profit if she filled her truck with the cargo from question 8?

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Name: Systems 2.3

Ready, Set, Go!

Ready Topic: Substitution Determine whether is a solution to each problem. 1. 2.

3. 4.

5.

Topic: Graphing two variable inequalities For each inequality and graph, pick a point and use it to determine which half-plane should be shaded, then shade the correct half-plane.

6.

7.

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Set Topic: Writing two variable inequalities Given the graph with the regions that are shaded write the inequality. 8. 9.

10. 11.

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Go Topic: Solve literal equations Re-write each of the following equations for the indicated variable. 12 for y 13. for x

14.

for d

Topic: Proportional relationships For each proportional relationship below, one representation is provided. Show the remaining representations and explain any connections you notice between representations.

15. Equation: Graph

Table

Days Cost 1 8 2 16 24

4 32

Create a Context

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16. Equation: Graph

Table

Create a Context

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2.4 Warm Up: Pampering and Feeding Time A Practice Understanding Task Carlos and Clarita have been worried about space and start‐up costs for their pet sitters business, but they realize they also have a limit on the amount of time they have for taking care of the animals they board. To keep things fair, they have agreed on the following time constraints.

Feeding Time: Carlos and Clarita estimate that cats will require 6 minutes twice a day— morning and evening—to feed and clean their litter boxes, for a total of 12 minutes per day for each cat. Dogs will require 10 minutes twice a day to feed and walk, for a total of 20 minutes per day for each dog. Carlos can spend up to 8 hours each day for the morning and evening feedings, but needs the middle of the day off for baseball practice and games.

Pampering Time: The twins plan to spend 16 minutes each day brushing and petting each cat, and 20 minutes each day bathing or playing with each dog. Clarita needs time off in the morning for swim team and evening for her art class, but she can spend up to 8 hours during the middle of the day to pamper and play with the pets.

Write inequalities for each of these additional time constraints. Shade the solution set that satisfies both constraints on the given coordinate grid.

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2.4 All For One, One For All A Solidify Understanding Task Carlos and Clarita have found a way to represent combinations of cats and dogs that satisfy each of their individual “Pet Sitter” constraints, but they realize that they need to find combinations that satisfy all of the constraints simultaneously. Why? 1. Begin by listing the system of inequalities you have written to represent the Startup Costs and Space

“Pet Sitter” constraints. (Section 2.1) 2. Find at least 5 combinations of cats and dogs that would satisfy both of the constraints represented by this

system of inequalities. List combinations below and plot on graph provided. How do you know these combinations work?

3. Find at least 5 combinations of cats and dogs that would satisfy one of the constraints, but not the other. List

combinations below and, in a new color, plot on graph provided. For each combination, explain how you know it works for one of the inequalities, but not for other?

4. Rewrite your systems of inequalities to include the additional constraints for Feeding Time and Pampering

Time. (Section 2.4) 5. Find at least 5 combinations of cats and dogs that would satisfy all four of the constraints represented by this

new system of inequalities. List combinations below and, in a new color, plot on graph provided. How do you know these combinations work?

6. Find at least 5 combinations of cats and dogs that would satisfy some of the constraints, but not all of them.

List combinations below and, in a new color, plot on graph provided. For each combination, explain how you know it works for some inequalities, but not for others?


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7. Using the provided resource page, shade the region in quadrant 1 that would represent all possible combinations of cats and dogs that satisfy the 4 “Pet Sitter” constraints. This set of points is referred to as the feasible region since Carlos and Clarita can feasibly board any of the combinations of cats and dogs represented by the points in this region without exceeding any of their constraints on time, money, or space.

8. Why is the feasible region restricted to quadrant 1 for this situation?

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Name: Systems 2.4

Ready, Set, Go!

Ready Topic: Solving Equations Determine the value of x that makes each equation true. 1. 2. 3.

Topic: Graphing two variable inequalities For each inequality and graph, pick a point and use it to determine which half-plane should be shaded, then shade the correct half-plane. 4. 5.

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Set Topic: Creating equations, solving real world problems, solve systems of equations A phone company offers a choice of three text-messaging plans. Plan A gives you unlimited text messages for $10 a month; Plan B gives you 60 text messages for $5 a month and then charges you $0.05 for each additional message; and Plan C has no monthly fee but charges you $0.10 per message. 6. Write an equation for the monthly cost of each of the three plans.

7. If you send 30 messages per month, which plan is cheapest?

8. What is the cost of each of the three plans if you send 50 messages per month? 9. Determine the values for which each plan is the cheapest? Topic: Solutions to systems of linear inequalities 10. Shade the solution to the following system on the graph below:

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Go Topic: Solve literal equations Re-write each of the following equations for the indicated variable.

11. for y 12.

for F

13. for b

Topic: Proportional relationships For each proportional relationship below, one representation is provided. Show the remaining representations and explain any connections you notice between representations.

14. Equation: Graph

Table

Create a Context Claire earns $9 per week allowance

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15. Equation: Graph

Table

Create a Context

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2.5 Warm Up Tickets Tickets Tickets to a show cost $10 in advance and $15 at the door. If 120 tickets are sold for a total of $1390, how many of the tickets were bought in advance? Assign variables and create a system of equations that models this situation. Solve the system algebraically using your preferred method. Graph the system to verify your solution.

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2.5 Get to the Point! A Solidify Understanding Task Carlos and Clarita need to clean the storage shed where they plan to board the pets. They have decided to hire a company to clean the windows. After collecting the following information, they have come to you for help deciding which window cleaning company they should hire.

Sunshine Express Window Cleaners charges $50 for each service call, plus $10 per window.

“Pane”less Window Cleaners charges $25 for each service call, plus $15 per window. 1. Which company would you recommend, and why? Prepare an argument to convince Carlos and Clarita

that your recommendation is reasonable. (It is always more convincing if you can support your claim in multiple ways. How might you support your recommendation using a table? A graph? Algebra?)

Number of Windows

Sunshine total charge

“Pane”less total charge

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Your presentation to Carlos reminds him of something he has been thinking about—how to find the coordinates of the points where the boundary lines in the “Pet Sitter” constraints intersect. He would like to do this algebraically since he thinks guessing the coordinates from a graph might be less accurate. 2. Write equations for the following two constraints (from section 2.1).

a. Space:

b. Startup Costs: Find where the two lines intersect algebraically. Record enough steps so that someone else can follow your strategy. 3. Now find the point of intersection for the two time constraints (from section 2.4).

a. Feeding Time:

b. Pampering Time: 4. What do the solutions for questions 2 and 3 represent in terms of the context of the “Pet Sitters” problem?

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Name: Systems 2.5

Ready, Set, Go!

Ready Topic: Determine patterns Find the next two values in the pattern. Describe how you determined these values. 1. 3, 6, 9, 12, ________, ________ Description: 2. 3, 6, 12, 24, ________, ________ Description: 3. 24, 20, 16, 12, ______, _______ Description: 4. 24, 12, 6, 3, ___ ___, _______ Description:

Set Topic: Solve systems of equations using substitution For questions 5-8 solve the system of equations using substitution. Check your work by graphing.

5. {

6. {

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7 {

8 {

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Go Topic: Graph two variable inequalities Graph the following inequalities.

9. 10. 11.

12. 13. 14.

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2.6 Warm Up Graphing linear inequality from a context Peggy is buying a balloon bouquet. Her favorite colors are silver and purple. The silver balloons are $1 and the purple balloons are $0.80. Graph an inequality that shows how many of each color balloon she can put in her bouquet if she doesn’t spend more than $20.

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2.6 Shopping for Cats and Dogs A Develop Understanding Task Clarita is upset with Carlos because he has been buying cat and dog food without recording the price of each type of food in their accounting records. Instead, Carlos has just recorded the total price of each purchase, even though the total cost includes more than one type of food. Carlos is now trying to figure out the price of each type of food by reviewing some recent purchases. See if you can help him figure out the cost of particular items for each purchase, and be prepared to explain your reasoning to Carlos. For questions 1 and 2, start by reasoning out the answer to each question by carefully examining the scenario. Record the thought process you used when reasoning out the answer to each question. Then write a system of equations and solve the system algebraically. 1. One week Carlos bought 3 bags of Tabitha Tidbits and 4 bags of Figaro Flakes for $43.00. The next week he

bought 3 bags of Tabitha Tidbits and 6 bags of Figaro Flakes for $54.00. Based on this information, figure out the price of one bag of each type of cat food. Explain your reasoning.

2. One week Carlos bought 2 bags of Brutus Bites and 3 bags of Lucky Licks for $42.50. The next week he bought 5

bags of Brutus Bites and 6 bags of Lucky Licks for $94.25. Based on this information, figure out the price of one bag of each type of dog food. Explain your reasoning.

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For questions 3-5, use the method of your choice (reasoning it out or solving algebraically) for solving each scenario. Carefully record your work. 3. Carlos purchased 6 dog leashes and 6 cat brushes for $45.00 for Clarita to use while pampering the pets. Later

in the summer he purchased 3 additional dog leashes and 2 cat brushes for $19.00. Based on this information, figure out the price of each item. Explain your reasoning.

4. One week Carlos bought 2 packages of dog bones and 4 packages of cat treats for $18.50. Because the finicky

cats didn’t like the cat treats, the next week Carlos returned 3 unopened packages of cat treats and bought 2 more packages of dog bones. After being refunded for the cat treats, Carlos only had to pay $1.00 for his purchase. Based on this information, figure out the price of each item. Explain your reasoning.

5. Carlos has noticed that because each of his purchases have been somewhat similar, it has been easy to figure

out the cost of each item. However, his last set of receipts has him puzzled. One week he tried out cheaper brands of cat and dog food. On Monday he purchased 3 small bags of cat food and 5 small bags of dog food for $22.75. Because he went through the small bags quite quickly, he had to return to the store on Thursday to buy 2 more small bags of cat food and 3 more small bags of dog food, which cost him $14.25. Based on this information, figure out the price of each bag of the cheaper cat and dog food. Explain your reasoning.

Summarize the strategies you have used to reason about the price of individual items in the problems given above. What are some key ideas that seem helpful?

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Name: Systems 2.6

Ready, Set, Go!

Ready Topic: Exponents Write the following in exponential notation. 1. 2.

Find each value.

3. 4. 5. 6. (

)

7. (

)

Write the following expression three different ways (one way can include the simplified value). 8. 9.

Set Topic: Solving systems 10. Nadia and Peter visit the candy store. Nadia buys three candy bars and four fruit roll-ups for $2.84. Peter also

buys three candy bars, but can only afford one additional fruit roll-up. His purchase costs $1.79. What is the cost of a candy bar and a fruit roll-up individually?

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11. A farmer noticed that his chickens were loose and were running around with the cows in the cow pen. He quickly counted 100 heads and 270 legs. How many chickens did he have and how many cows?

12. Solve the system of equations created by one of the following combinations of “Pet Sitters” constraints using

the elimination of variables method. a. Space and Feeding Time b. Space and Pampering Time c. Start-up Costs and Feeding Time d. Start-up Costs and Pampering Time

Topic: Solve systems of equations Solve the following systems of equations using elimination of variables, then justify graphically.

13. {

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14. {

Go Topic: Solve one variable inequalities. Solve the following inequalities. Graph the solution set on a number line.

15. –

16. (

)

50


17. (

)

Solve each inequality. Give the solution in inequality notation.

18.

19.

20. 21.

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2.7 Warm Up Profiting Brian asked his friend Karim for advice about what combination of pictures to pain. He suggested that Brian determine a reasonable profit for that month’s work. Then he could paint what he needs in order to earn that amount of profit. Here are the facts about Brian’s paintings.

Each acrylic painting requires $20 in materials and earns a profit of $120. Each pastel art piece requires $4 in materials and earns a profit of $50. Brian has $180 to spend on materials. Brian can paint at most 21 pictures.

See if you can help Brian and Karim. Prepare a written report on the situation. This report should include your work on questions 1-4. Most importantly, it should contain your explanation for question 5. 1. You have already found the feasible region for the problem (see Profiting from Pictures). Remember, the

feasible region is the set of points that satisfy the constraints. Copy this feasible region to use in this activity. Label the axes and show the scales.

2. Suppose Brian decides that $1300 would be a reasonable profit. a. Find three combinations of acrylic paintings and pastels that would earn Brian a profit of exactly $1300.

b. Mark these three number pairs on your graph.

3. Now suppose Brian wants to earn only $600 in profit. Find three combinations of acrylic paintings and pastels

that will earn a profit of exactly $600. Using a different color or symbol, add these points to your graph.

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4. Now suppose Brian wants to earn only $800 in profit. Find three combinations of acrylic paintings and pastels that will earn a profit of exactly $800. Using a third color or symbol, add these points to your graph.

5. Brian’s mother has convinced him that he should try to earn as much as possible. Brian wants to figure out the

highest profit he can make within his constraints. He also wants to prove to his mother than this profit really is the maximum amount. a. Find the maximum possible profit that Brian can earn and the combination of picture he needs to paint to

earn that profit. b. Write an explanation that will convince Brian’s mother that your answer is correct.

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2.7 Food for Fido and Fluffy A Solidify Understanding Task Carlos and Clarita have found two different cat foods that seem to appeal to even the most finicky of cats: Tabitha Tidbits and Figaro Flakes. Each ounce of Tabitha Tidbits contains 2 grams of protein, 4 grams of carbohydrates and 4 grams of fat. Each ounce of Figaro Flakes contains 3 grams of protein, 4 grams of carbohydrates and 2 grams of fat. Since Tabitha Tidbits is fairly expensive, while Figaro Flakes is very cheap, the twins have decided to create a new cat food by mixing the two. After studying some nutritional guidelines for cats, Carlos and Clarita have decided to create a mixture based on the following constraints.

• Amount of Protein: Each meal should contain at least 12 grams of protein. • Amount of Carbohydrates: Each meal should contain more than 16 grams of carbohydrates. • Amount of Fats: Each meal should contain no more than 18 grams of fat. • Size of a Feeding: Each meal should consist of less than 10 ounces of food.

For the work that follows, let T represent the number of ounces of Tabitha Tidbits in a meal and let F represent the number of ounces of Figaro Flakes. 1. Write an inequality for each of the constraints. 2. Graph the solution set for each of the inequalities you wrote in #1. How do you know on which side of the

boundary line you should shade the half‐plane that represents the solution set?

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3. Decide if the boundary line for each inequality represented in #2 should be a solid line or a dotted line. Which words or phrases in the constraints suggested a solid line? A dotted line?

4. Find at least 5 combinations of Tabitha Tidbits and Figaro Flakes Carlos and Clarita can mix together to create a

nutritious cat meal. Show on your graph above that these points lie within a feasible region for the constraints. 5. Find the corner points of the feasible region algebraically. 6. Brutus Bites is a brand of dog food that contains 4 grams of protein and 6 grams of fat per ounce. Lucky Licks is

another brand of dog food that contains 12 grams of protein and 4 grams of fat per ounce. Carlos wants to make a meal for dogs that contains at least 8 grams of protein and no more than 6 grams of fat. Write and solve a system of inequalities that Carlos can use to determine a combination of Brutus Bites and Lucky Licks that will satisfy these constraints.

7. Find the corner points of the feasible region algebraically.

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Name: Systems 2.7

Set, Go!

Set Topic: Writing equations of two variable inequalities Given the graph with the regions that are shaded write the inequality or system of inequalities. 1. 2. 3.

Go Topic: Graph two variable inequalities Graph each set of inequalities below. Include the shaded region of both, plus indicate the region that is true for all inequalities.

4. {

5. {

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Solve the following systems of equations.

6. {

7. {

Graph each inequality on the coordinate plane provided.

8. 9.

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In Or Out For each of the systems of inequalities, determine if the given coordinates are solutions to the system.

1. {

a. b. c.

2. {

a. b. c.

3. Graph the following system of inequalities and list 3 points that are solutions to the system.

{

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2.8 Taken Out of Context A Practice Understanding Task Write a shopping scenario similar to those in “Shopping for Cats and Dogs” to fit each of the following systems of equations. Then use the elimination of variables method to solve the system. Some of the systems may have interesting or unusual solutions. See if you can explain them in terms of the shopping scenarios you wrote.

1. {

2. {

3. {

© 2

01

2 w

ww

.fli

ckr.

com

/ph

oto

s/m

om

mav

en

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4. {

5. Extend your thinking…Three of Carlos’ and Clarita’s friends are purchasing school supplies at the bookstore.

Stan buys a notebook, three packages of pencils and two markers for $7.50. Jan buys two notebooks, six packages of pencils and five markers for $15.50. Fran buys a notebook, two packages of pencils and two markers for $6.25. How much do each of these three items cost?

Explain in words or with symbols how you can use your intuitive reasoning about these purchases to find the price of each item.

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Name: Systems 2.8

Set, Go!

Set Topic: Determine the number of solutions in a system of equations Express each equation in slope-intercept form. Without solving, state whether the system of equations has zero, one, or infinite solutions. Justify your response.

1. {

2. {

3. {

4. {

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Go Topic: Graph two variable inequalities Graph the following inequalities. Be sure to label your axes and scale. Justify the region you shade by showing three points in the region as being solutions to the problem. Show a point you have tested to prove your shaded region is accurate.

5. 6

7. 8.

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9. On the same set of axes, graph and . What values do these two have in common?

Topic: Solving systems of inequalities Solve each system of inequalities. Be sure to darken the solution set on the graph. 10. 11.

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2.9 Warm Up Profiting from Hens and Abacuses Remember Tamara and her truck hauling Heidi’s Hen Houses and Amy’s Abacuses? It’s now time to help Tamara maximize her profit from hauling cargo up and down the state during her free time. Here is a reminder about Tamara’s trucking business:

Heidi’s Hen Houses weigh 40 pounds, are 3 cubic feet in volume, and make a profit of $4 per container. Amy’s Abacuses weigh 30 pounds, are 4 cubic feet in volume, and make a profit of $2.50 per container. Tamara’s truck can hold a maximum of 2800 cubic feet. Tamara’s truck can carry a maximum weight of 31,500 pounds.

1. You have already found the feasible region for the problem (see Tamara’s Trucks). Remember, the feasible

region is the set of points that satisfy the constraints. Copy this feasible region to use in this activity. Label the axes and show the scales.

2. Suppose Tamara decides that $1000 would be a reasonable profit. a. Find three combinations of products that would earn Tamara a profit of exactly $1000.

b. Mark these three number pairs on your graph.

3. Now suppose Tamara wants to earn only $2000 in profit. Find three combinations of products that will earn a

profit of exactly $2000. Using a different color or symbol, add these points to your graph.

4. Tamara wants to figure out the highest profit she can make within her constraints. Find the maximum possible profit that Tamara can earn and the combination of products she needs to haul to earn that profit.

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2.9 Pet Sitters Revisited A Develop Understanding Task Carlos and Clarita have successfully found a way to represent all of the combinations of cats and dogs that they can board based on all of the following constraints.


Feeding Time: Carlos and Clarita estimate that cats will require 6 minutes twice a day— morning and evening—to feed and clean their litter boxes, for a total of 12 minutes per day for each cat. Dogs will require 10 minutes twice a day to feed and walk, for a total of 20 minutes per day for each dog. Carlos can spend up to 8 hours each day for the morning and evening feedings, but needs the middle of the day off for baseball practice and games.

Pampering Time: The twins plan to spend 16 minutes each day brushing and petting each cat, and 20 minutes each day bathing or playing with each dog. Clarita needs time off in the morning for swim team and evening for her art class, but she can spend up to 8 hours during the middle of the day to pamper and play with the pets.


Now they are trying to determine how many of each type of pet they should plan to accommodate. Of course, Carlos and Clarita want to make as much money as possible from their business, so they need to pay attention to both their daily income as well as their daily costs. They plan to charge $8 per day for boarding each cat and $14 per day for each dog. They estimate that each cat will require $2.00 per day in food and supplies, and that each dog will require $4.00 per day in costs. After surveying the community regarding the pet boarding needs, Carlos and Clarita are confident that they can keep all of their boarding spaces filled for the summer. So the question is: How many of each type of pet should they prepare for in order to make as much money as possible?

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To get started on this task, you might want to look for collections of points where the daily profit is the same. For example, can you find a collection of points where, for each point, the daily profit is $120? Plot these points on the provided graph. What do you notice? Do the same for a profit of $150 and $180. What do you notice? What combination of cats and dogs do you think will make the most money? What recommendations would you give to Carlos and Clarita, and what argument would you use to convince them that your recommendation is reasonable?

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Name: Systems 2.9

Ready, Go!

Ready Topic: Solve exponential equations Find the value of x for each situation. 1. 2. 3. 4.

Go Topic: Find the solution region of the following systems of inequalities. 5. Write the system of inequalities that is represented in the graph to the

right. Graph each set of inequalities and determine the solution region.

6. {

7. {

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8. {

9. {

Topic: Solving systems of equations by substitution. Solve each system using substitution. 10. 11.

Topic: Solving systems of equations by elimination. Solve each system using elimination. 12. 13.

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2.10 Warm Up Reflecting on Carlos and Clarita Describe the module’s central problem and main mathematical ideas. Your description should give an overview of how the key ideas were developed. Also describe how you used them to solve the central problem.

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Name: Systems 2.10

Go!

Go Topic: Solving systems by graphing For #1-3, solve each system of equation by graphing: 1.

2.

3.

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Topic: Solving systems by substitution & elimination For #4-6, solve each system of equations using the substitution: 4.

5.

6.

For #7-10, solve each system by elimination: 7.

8.

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9.

10.

Topic: Systems of equatiosn word problems 11. When not playing volleyball, the team decided to rent bicycles. They learned that the cost to rent a bike

included an initial fee of $25 and an hourly rate of $5 per hour. Identify the starting point: _________ Identify the slope: ________.

a. Write an equation that relates the total cost (c) to the number of hours (h).

b. Use your equation to compute the total cost of a 6 hour rental.

c. How would your equation change if the team rented 5 bicycles?

d. Use your equation in part c to compute the total cost of a 6.5 hour rental.

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12. You work for a small business that sells bicycles, tricycles, and tandem bikes. Bicycles have one seat, two pedals, and two wheels. Tricycles have one seat, two pedals, and three wheels. Tandem bikes have two seats, four pedals, and two wheels.

a. On Monday you counted 48 tricycle wheels. How many tricycles were in the shop?

b. Write an algebraic equation that shows the relationship between the number of wheels (w) and the number of tricycles (t).

c. On Wednesday there were no tandem bikes in the shop. There were only bicycles and tricycles. There are a total of 24 seats and 61 wheels in the shop. How many bicycles and how many tricycles are in the shop? Show how you figured it out using algebra.

Topic: Solving systems of inequalities For #13-14, solve each system of inequalities. 13.

14.

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15. Jason is buying wings and hot dogs for a party. One package of wings costs $7. Hot dogs cost $4 per pound. He must spend less than $40. a. Write an inequality to represent the cost of Jason’s food for the party. b. Jason knows that he will be buying at least 5 pounds of hot dogs. Write an inequality to represent this

situation. c. Graph both inequalities and shade the intersection.

d. Identify two solutions and justify your answers.

Integrated Math 1 Module 2 Systems of Equations and...

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